LOCAL LINEAR FITTING UNDER NEAR EPOCH DEPENDENCE: UNIFORM CONSISTENCY WITH CONVERGENCE RATES

Degui Li,Zudi Lu,Oliver Linton

doi:10.1017/s0266466612000011

Abstract

Local linear fitting is a popular nonparametric method in statistical and econometric modeling. Lu and Linton (2007, Econometric Theory23, 37–70) established the pointwise asymptotic distribution for the local linear estimator of a nonparametric regression function under the condition of near epoch dependence. In this paper, we further investigate the uniform consistency of this estimator. The uniform strong and weak consistencies with convergence rates for the local linear fitting are established under mild conditions. Furthermore, general results regarding uniform convergence rates for nonparametric kernel-based estimators are provided. The results of this paper will be of wide potential interest in time series semiparametric modeling.

Highlights

Local linear ...tting is a popular nonparametric method in nonlinear statistical and econometric modelling
Lu and Linton (2007) recently established the pointwise asymptotic distribution for the local linear estimator of a nonparametric regression function under the weak assumption of near epoch dependence, which cover a wide range of popular time series econometric models
We further investigate the uniform consistency of this nonparametric estimator for near epoch dependent processes

Summary

Introduction

Local linear ...tting is a popular nonparametric method in nonlinear statistical and econometric modelling. We discuss the application of the obtained results to establish the uniform convergence rate of local linear estimator of conditional variance function. Before the end of this section, we remark that some alternative extension of dependence beyond mixing can be found in Nze, Bühlmann, and Doukhan (2002) and Nze and Doukhan (2004) who investigated a class of dependent processes they call “weak dependent", the de...nition of which is quite involved They established the asymptotic normality and uniform convergence of local constant nonparametric regression estimators under their conditions. There has been rich literature on the uniform convergence rate for the local linear estimators under mixing dependent condition, see Masry (1995), Fan and Yao (2003) and Hansen (2008) for example. Suppose that the conditions in Theorem 2.1 are satis...ed, EjYtjs < 1 and EkXtks < 1, s > 2p0,

General uniform convergence

Applications

Estimation of a countable number of conditional expectations

Linear process